During my first fall of teaching in 2003, the school district I worked for passed a bond issue to build a new high school. The following spring the architects and facility committee (which I would later join) asked for staff input. So I ask you: What would your ideal school look like? What features would you prioritize? How would you maximize utility within a budget?

I was reminded of this activity by a recent post by Zac Chase. In that post, he discusses a class activity where he and other students plan the layout of a school in order to think about the environments that best support teaching and learning. In 2004, when I was faced with a similar task, I set to work sketching on paper. When sketching wasn't good enough, I went with graph paper and scale drawings. When graph paper wasn't good enough, I downloaded a free CAD program, taught myself how to use it, and set out to design an entire school. I was a bit obsessed. I had the following goals and guidelines to consider:

The design needed to accommodate 750 students in classrooms and have "common spaces" (gym, locker room, cafeteria, library, etc.) to handle an expansion up to 1500 students.

The school budget allowed for approximately 115,000 square feet.

Hallways are an expensive use of little-used space. I wanted to minimize them.

Departments would have common computer lab space and office space for teachers.

After a lot of editing and calculating, this is what I came up with on the first floor (open link in new tab and zoom for maximum detail):

First floor

And here was the plan for the second floor:

Second floor

I was (and still am) pretty proud of this creation. In fact, I still believe this school offers a better use of space than the school that was eventually built. However, with 7+ years of hindsight, I think this plan could have used the following improvements:

Instead of shared "computer lab" space for each department, students would take their technology with them to the classroom.

In place of the computer spaces, I'd provide more space for student group collaboration. I was thinking about the office for teacher collaboration space, but at the time I didn't think enough about student collaboration space.

I didn't think too much about the cafeteria, and it shows. Dining areas should be more inviting than the rows-of-tables design I designed.

One other lesson I learned after serving on two facility committees is this: Don't just ask teachers what they want in their classroom and school. Ask teachers what they imagine a teacher twenty years from now will want. Otherwise, it's too easy to give teachers the impression that the school is being custom-built for them, something that's not likely to happen. It might not even be desirable. Instead, ask them questions that reflect the school's long-term value to the community.

Those are important, but otherwise I think my floorplans still have something to offer. I don't know why I hadn't shared them until now, but maybe they'll be of some use to somebody looking for school building ideas. Just promise me that if you build this thing, invite me to come see it!

My officemate, Ryan Grover, had put it there to tease us with what I quickly believed was a paradox. Having dealt with paradoxes before, I knew the first rule of paradoxes: "Don't try to reason with paradoxes." Instead, I thought I'd post it to Google+ to see if some of my math-oriented followers could have some fun with it.

Ryan will be the first to admit that he is not the creator of this problem. He told me that he had seen something like it, had given it some thought, and then searched for similar problems. Ryan found variationsof the problemon Reddit, got his wording just right, and wrote it on the chalkboard. He knew it was worth sharing, just as I did. But I had a different sharing mechanism in mind.

I posted on October 20th and nothing much happened the first few days. Then things really took off. You can get a sense for some of the progress thanks to Google+ Ripples. But what really got my attention was seeing Brian Brushwoodshare the picture on October 27th. I follow Brian because of his work on TWiT.tv, not because I expect him to post interesting probability problems. Because the picture had made its way to Twitter, where Brian saw it, and then back to Google+, it was no longer connected to my original post. Still, Brian's post quickly got over 2000 reshares -- including one by Terence Tao. I argued to Ryan's (and my) advisor that "getting cited" by Terence Tao should satisfy Ryan's "publishable work" requirement of our PhD program, but I don't think he bought it.

The same day Brian posted the picture it hit Reddit (and againin daysfollowing), getting over 2300 comments, probably 10 times more than in previous posts. I wonder if it's the placement of the problem on the chalkboard that added to its appeal, or if it was just the right question at the right time to engage people's interest. The next day the picture appeared again in Google+, this time in Ed Yong's stream where it got over 1500 reshares and 300 comments. It was also quite a treat to see the picture show up at FlowingData.com. I'm a pretty big fan of Nathan Yau's blog and book, and in an email Nathan said the 788 comments were probably the longest thread of any post ever made on FlowingData.

There's no way to count precisely, but the picture was reshared publicly on Google+ (where an estimated two-thirds of posts are private) around 5000 times and comments on Google+, Reddit, and FlowingData also number in the several thousand. Some good things have come of this:

It's been fascinating to watch people try to reason through the problems in the comments. Math teachers like to watch people who don't give up easily.

I've picked up several hundred followers on Google+ from all over the world. Many of them have interests in mathematics and how it is taught and learned, and what they've shared with me is many times more valuable than even seeing the comments and shares from Brushwood, Tao, and Yau.

It's bolstered my (and my colleagues') belief that if something is interesting, it should be shared. We're not in the business of keeping good ideas to ourselves.

As far as I can tell, maybe only two bad things have come from this:

We've experienced a little bit of "sharer's guilt" because neither Ryan or I deserve any credit for actually coming up with the problem. Someone who shared the problem before we did, such as in those previous Reddit posts, might be feeling justifiably peeved that they aren't getting the credit they're due. We're sorry.

We've avoided erasing that portion of our chalkboard, even though we could use the space. Next time Ryan has a perplexing problem he'll have to write smaller. :)

Many math education researchers come from one of two camps: (a) math teachers who want to know more about the psychology of the student, or (b) psychologists who want to know more about how students learn math. When these groups of researchers work together, good things can happen.

Previous research had indicated two key ideas: (a) a proper understanding of equals and equivalence is key to success in algebra, and (b) the equal sign and equivalence is misunderstood by students at all ages. While some of this previous research is very good in its own right, the longitudinal aspect of this study helps it stand out. Commonly, and incorrectly, students hold an operational view of the equals sign. To those students, they see the "=" sign as meaning "do something." When students encounter a problem like \(3+5=\) they think the equal sign is a prompt to "write the answer," which in this case is 8. Unfortunately, some of those same students will see a problem like \(3+5=x+2\) and still think \(x=8\). Not knowing what to do with the 2, they might also think \(x=10\) (because they're just adding all the numbers they see) or they think it's okay to write \(3+5=8+2=10\). Statements like this with multiple equal signs should look familiar to any math teacher who has watched students show their work for a multi-step problem, such as showing work for order of operations. This only makes sense to students who have an operational view of the equal sign, as to them it just means, "the answer to this step is." But that's incorrect. Instead, we want students to understand the equal sign as a relational symbol, one that is neither prompting action nor implying a direction to that action. Without this, solving equations in algebra has very little meaning.

For this part of their study, Alibali et al. studied 81 middle school students (62% white, 25% African American, 7% Asian, 5% Hispanic) from 6th grade through 8th grade. The middle school used the Connected Mathematics curriculum and introduced solving linear equations in grade 7. The students were asked to explain what they thought the "=" sign meant, and to understand their use of that sign they were given an interesting set of tasks. For example:

Is the value of n the same in the following two equations? Explain.

\( 2 \times n + 15 = 31 \) and \(2 \times n + 15 -9 = 31 - 9\)

Here the researchers apply what they call an "atypical transformation," and they look carefully at how students find n. Many students would solve by "doing the same thing to both sides" for both equations, a procedure they can follow whether they had a solid understanding of equals and equivalence or not. But by subtracting 9 from each side in the second equation -- something mathematically "legal" despite not being all that helpful in finding n -- you can more easily identify which students break with "standard" procedure and show an understanding of equivalence. Those students won't treat the second equation like a new problem and instead quickly see that whatever they found for n in the first equation must also be n in the second.

Not surprisingly, Alibali et al. found that students' understanding of the equal sign got better over time. Also not surprisingly, students who have the correct, relational view of equals are more likely to see equivalence relations and solve equations correctly, and the earlier they understand it, the better. At the beginning of 6th grade, about 70% of students had an operational view and only 20% had a relational view. (10% of students held some other view that didn't fit in these two conceptions of equals.) By the end of 8th grade, that balance had almost flipped: only about 30% still held an operational view while 60% had a relational view. That's a lot of improvement, but that improvement took a long time (3 years) and still 40% of students didn't have a correct and meaningful understanding of the equals sign by the end of 8th grade. Also, students who showed a relational view of equals sometimes slipped back into an operational view. Almost a quarter of the students in the study used a less sophisticated strategy sometime after using a better one. Lastly, even when students consistently defined the equal sign as a relational symbol, they didn't always recognize equivalence in problems such as the one above. It's these types of caveats that make teaching equals and equivalence a tricky business.

So if you're a teacher with students having trouble with the equal sign, what can you do? More research needs to be done in this area, but one thing you can do is be more aware of your students' "compulsion to calculate" (a clever term used by Stacey & MacGregor, 1990, p. 151, as cited by Alibali et al., 2007, p. 245). Try giving students a task like the one above, ask them to evaluate the task for a minute or two without touching their pencils, and then find n. Afterwards, have students describe their strategies and solutions. Also, if you want to avoid the complications of using a variable, you can give students a number of statements and see if they can spot the ones that are equivalent. (Alibali et al. suggest statements like 9 + 5 = 14, 9 + 5 - 3 = 14 - 3, and 9 + 5 - 3 = 14 + 3). Also, try putting the unknown to the left of the equals sign. If you ask a student to solve \( \underline{\hspace{0.25in}} = 3 + 4 \) and they tell you the problem is "backwards," then you know they struggling with an operational view of equals. Giving those students more problems where the "answer" doesn't come "last" (to the right or at the bottom) will help the student expand their understanding of what equals really means.

This is the fourth in a series of posts describing "Research You Should Know" (RYSK). While the article is not actually the report of research findings, it is part of a foundation upon which a generation of mathematics education research has been based.

Starting in the late 1960s, Dutch mathematician Hans Freudenthal saw the trend of "new math" spreading from the U.S. to the world. He pushed back with a philosophy of mathematics education now known as Realistic Mathematics Education (RME). The following article by Freudenthal, Why To Teach Mathematics So As To Be Useful, provides early insight to the core principles of RME: mathematics as a human activity, mathematization from contexts, and mathematics for all students. This article is also the first article in the first ever issue of the journal Educational Studies in Mathematics. Thankfully, instead of simply summarizing the article, I've been granted permission to reprint it here so you can read Freudenthal's words for yourself.

HANS FREUDENTHAL

WHY TO TEACH MATHEMATICS

SO AS TO BE USEFUL

My first task at this moment is to welcome you who have come here from various countries to sacrifice one week of your holidays for the benefit of mathematical education all over the world. I trust this meeting will be as useful as according to the general theme of this conference mathematical education should be held to be. I trust we all will learn as much from each other's experiences and arguments as we like to do and often have done at such opportunities. With great satisfaction I remember the meeting of December 1964 at Utrecht and I hope the few among you who have participated in that conference will share my feelings of gratitude. But whenever I shall remember those pleasant days and evenings, and lively discussions, I will never forget the man whom I met first and last on that occasion, the liveliest among all of us, the much regretted Wittenberg, this fiery nature who died much too early as though he had burnt himself in his own fire. Though I admit there was none among us who shared his opinions, I am sure everybody was impressed by his honest search into the truth of our educational philosophy. To my mind, his definitive absence overclouds the bright sky of this day.

The present colloquium is an activity of the ICMI sponsored by the government of the Netherlands and by IMU. It is not the first in this new period of ICMI and in this year. In January we met in Lausanne with the physicists, in a meeting sponsored by UNESCO, which was attended by some among you. In my opinion the resolutions adopted at Lausanne are a mile-stone in the philosophy of mathematical education. If I substitute my wishes and hopes for my opinion, I would say they should be so. It is evident that the use of mathematics has been a key criterion in all arguments on mathematics at that meeting.

In this introductory address I feel I have to justify the general theme of the present conference rather than to tell you about techniques of teaching useful mathematics. This means that I will not speak about how to teach mathematics so as to be useful but about why we should teach mathematics so as to be useful, or rather about why we should teach mathematics so as to be more useful.

Of course this is a question of educational philosophy, and as such it will be answered in a different way according to which philosophy we adhere to. Yet educational philosophy is not an abstract system. It depends on the real educational system in which we live, and on our, positive or negative, attitude with respect to that system. Is the variety of national educational philosophies really a drawback to international talks on mathematical education or should I say that there is no better opportunity to test them than to have them bump against each other? Are not we too often and too readily inclined, when reading or hearing about the educational experiences in another country, under another educational system, to sigh: it is just a pity, but this does not apply to our situation? I would say whenever this happens, then something is wrong either in the one system or the other, or, most likely, in both.

It is generally admitted that there is a wide gap between the educational philosophy of the U.S.A. and the Socialist European countries on one hand and the continental Western European countries on the other hand, though this gap has been narrowing to a considerable extent. On the one side one has for long times pursued the ideal of one kind of education for all youth, on the other side one has always overstressed that part of the educational system which provides educational facilities for a small group of students selected more on social than on intellectual grounds. I have to admit, and I do it with shame and distress, that in the Western countries of continental Europe, if we speak about mathematical education, we more often than not, mean the gymnasiums and lycées, and tacitly forget the about more than 90% who do not attend this type of schools. I agree that a more balanced educational system can be as bad if its highest level is too low to do justice to the most gifted students. But instead of discussing the question which kind of justice is the least evil, I would rather try to do the most justice to all people and to the society they belong to.

I need not explain to you why mathematics can be useful though the fact itself is one of the most recent and most astonishing features of the history of civilization. It would be more difficult to tell how mathematics can be useful provided that we do not limit ourselves to counting up instances of the all-pervading influence of mathematics in our culture, but ask what happens in the individual if he applies mathematics or if he tries to. Much has been done to investigate the learning process, though it is a fact that most of this research has been rather laboratory than classroom-oriented. Very little, if anything, is known about how the individual manages to apply what he has learned, though such a knowledge would be the key to understanding why most people never succeed in putting their theoretical knowledge to practical use.

Since mathematics has proved indispensable for the understanding and the technological control not only of the physical world but also of the social structure, we can no longer keep silent about teaching mathematics so as to be useful. In educational philosophies of the past, mathematics often figures as the paragon of a disinterested science. No doubt it still is, but we can no longer afford to stress this point if this keeps our attention off the widespread use of mathematics and the fact that mathematics is needed not by a few people, but virtually by everybody.

Mathematics is distinguished from other teaching subjects by the fact that, even in its actual totality, it is a comparatively small body of knowledge, of such a generality that it applies to a richer variety of situations than any other teaching subject. Modern mathematics can be seen as an effort to reduce this body of knowledge even more and to enhance the flexibility of what remains to be taught. At the same time this fact about mathematics is the source of the principal dilemma in teaching mathematics so as to be useful. In an objective sense the most abstract mathematics is without a doubt the most flexible. In an objective sense, but not subjectively, since it is wasted on individuals who are not able to avail themselves of this flexibility. On the other hand, teaching applied mathematics is as bad, if it means mathematics in a specialized context, which does not account for the greatest virtue of mathematics, its flexibility.

Though it might look different, I am still busy with the question why mathematics has to be taught so as to be useful, after we had agreed that it is useful and that students are expected to use it. There are two extreme attitudes: to teach mathematics with no other relation to its use than the hope that students will be able to apply it whenever they need to. If anything, this hope has proved idle. The huge majority of students are not able to apply their mathematical classroom experiences, neither in the physics or chemistry school laboratory nor in the most trivial situations of daily life. The opposite attitude would be to teach useful mathematics. It has not been tried too often, and you understand that this is not what I mean when speaking about mathematics being taught to be useful. The disadvantage of useful mathematics is that it may prove useful as long as the context does not change, and not a bit longer, and this is just the contrary of what true mathematics should be. Indeed it is the marvellous power of mathematics to eliminate the context, and to put the remainder into a mathematical form in which it can be used time and again.

Between two extreme attitudes one may be inclined to try compromising. If this means teaching pure mathematics and afterwards to show how to apply it, I am afraid we are no better off. I think this is just the wrong order. I have always considered it a remarkable fact that people are able to apply simple arithmetic, but not quadratic equations or even linear functions. Do not object that arithmetic is so easy. It is not. Take such problems as:

If I have got ten marbles and I give three away, how many are left?
If I have got ten marbles, and John has three less, how many does he have?
If there are ten students in the room and three are girls, how many are boys?
If I am ten years old now, how old was I three years ago?
If B is between A and C, B is at a distance of 7 miles from A, and C is at a distance of 10 miles from A, how far is B from C?

It is not so easy to learn that in all these and a hundred other situations the same arithmetical operation applies. It takes some time, but finally everybody succeeds in understanding it. Why? I daresay, because arithmetic starts in a concrete context and patiently returns to concrete contexts as often as needed. The counterexample is fractions. In its traditional teaching the concrete context is no more than a ceremony which is hurried through in a jiffy. If afterwards the abstract theory of fractions has to be applied, its comes too late, on too high a level, and is not connected to any previous experience on a level where fractions should have been introduced. What is the reason for this change of attitude of the teacher? Is the patience of the schoolmaster exhausted when fractions turn up? I believe the answer is rather that the schoolmaster himself does not know fractions in a concrete context, and that for this reason he is not able to teach them in a more responsible way than he is used to do.

I am afraid this answer applies to the greater part of our mathematics teaching. Even the fact that a teacher applies mathematics himself, does not necessarily imply that he knows how he is able to do so and to use such a knowledge in his teaching.

The problem, however, is still much more serious. In the past, and mostly even now, textbook writing has been dominated by quite other aims than by the goal of a mathematics that could be useful. Mathematics is a peculiar subject. Arithmetic and geometry have sprung from mathematizing part of reality. But soon, at least from the Greek antiquity onwards, mathematics itself has become the object of mathematizing. Arranging and rearranging the subject matter, turning definitions into theorems and theorems into definitions, looking for more general approaches from which all can be derived by specialization, unifying several theories into one -- this has been a most fruitful activity of the mathematician, and no doubt our students are entitled to enjoy these fruits. No doubt modern mathematics is both much more flexible and much simpler than the mathematics of fifty years ago. No doubt our students have to learn the most modern mathematics. Teachers are more and more prepared and more and more inclined to bridge the gap between school mathematics and grown-up mathematics which had become wider from year to year.

However, this is not the whole story. The problem is not what kind of mathematics, but how mathematics has to be taught. In its first principles mathematics means mathematizing reality, and for most of its users this is the final aspect of mathematics, too. For a few ones this activity extends to mathematizing mathematics itself. The result can be a paper, a treatise, a textbook. A systematic textbook is a thing of beauty, a joy for its author, who knows the secret of its architecture and who has the right to be proud of it. Look how such an author would justify his construction: Why have you defined addition on page 10 in such a circumstantial way? -- because this more general definition will prove useful on p. 110. Why have you proved this geometrical theorem in such an unnatural manner? -- because at this stage I restrict myself to affine notions which have to precede metric notions. Why do not you mention forces as an instance of vectors? -- because mechanics has to be based upon vector algebra and not the other way round.

Systematization is a great virtue of mathematics, and if possible, the student has to learn this virtue, too. But then I mean the activity of systematizing, not its result. Its result is a system, a beautiful closed system, closed, with no entrance and no exit. In its highest perfection it can even be handled by a machine. But for what can be performed by machines, we need no humans. What humans have to learn is not mathematics as a closed system, but rather as an activity, the process of mathematizing reality and if possible even that of mathematizing mathematics.

New mathematics has been met with criticism. People who apply mathematics often feel uneasy when observing that the mathematics they have been used to apply is replaced by something they judge less suited for applications. It is a fact that biologists, economists, sociologists are better prepared to apply modern mathematics than physicists who carry the burden of a longer tradition. In the universities the gap between the mathematics of mathematicians and that of physicists has become terrifying. It is a habit of physicists to treat any particular subject with that kind of mathematics which prevailed at the time when that subject turned up in the history of physics. For instance, though physicists know eigenvalues of symmetric matrices because Laplace introduced them in a physical context, they still deal with orthogonal matrices with such oddities as Eulerian angles, because Euler was not yet acquainted with eigenvalues.

It would be a disaster if this lag would become permanent, though I hope it will not. Time ago I eavesdropped on a talk between a physics professor and his assistants, criticizing his course and particularly such a subject as Lagrange multipliers: this is not physics, one of them said, this is plain linear algebra.

Probably we will have to wait for the next generation to have physicists reconciled with modern mathematics teaching.

It is a pity that most of the criticism against modern mathematics is made with no knowledge about what modern mathematics really is. It is a pity, because there is ample reason for such criticism as long as mathematicians care so little about how people can use mathematics. We are not entitled to reproach physicists for identifying modern mathematics with a preposterous educational philosophy, since this identification is of our own making. I am convinced that, if we do not succeed in teaching mathematics so as to be useful, users of mathematics will decide that mathematics is too important a teaching matter to be taught by the mathematics teacher. Of course this would be the end of all mathematical education.